$A$ motorcyclist wants to drive in horizontal circles on the vertical inner surface of a large cylindrical wooden well of radius $8.0 \ m$,with a minimum speed of $5 \sqrt{5} \ m \ s^{-1}$. The minimum value of the coefficient of friction between the tyres and the wall of the well must be (take $g = 10 \ m \ s^{-2}$):

The maximum safe speed of a car on a flat circular road of radius $4 \; m$ is $4.9 \; m/s$. What is the coefficient of friction between the road and the tires?

$A$ car of mass $1500 \ kg$ is moving with a speed of $12.5 \ m/s$ on a circular path of radius $20 \ m$ on a level road. What should be the coefficient of friction between the car and the road,so that the car does not slip?

$A$ body of mass $200 \text{ g}$ is tied to a spring of spring constant $12.5 \text{ N/m}$,while the other end of the spring is fixed at point '$O$'. If the body moves about '$O$' in a circular path on a smooth horizontal surface with a constant angular speed of $5 \text{ rad/s}$,then the ratio of the extension in the spring to its natural length will be:

The coefficient of static friction between the road and tyres of a car is $0.4$. The maximum permissible speed of the car is $10 \,ms^{-1}$ on a curved unbanked road. Then the maximum radius of curvature of the road is (acceleration due to gravity $= 10 \,ms^{-2}$)

$A$ coin placed on a rotating table just slips if it is placed at a distance $4r$ from the centre. On doubling the angular velocity of the table,the coin will just slip when at a distance from the centre equal to